It suffices to
show that there
exists a constant \(C>0\)  such that
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}B(x,r)}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{B(x,r)}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{B(x,r)}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{B(x,r)}} \operatorname{dist}(y,E)^{-\alpha }\,dy\le C \operatornamewithlimits{ess\, inf}_{y\in B(x,r)}\operatorname{dist}(y,E)^{-\alpha }\) 
for all
\(x\in E\)  and \(r>0\) .
Indeed, if \(\operatorname{dist}(Q,E)< 2\operatorname{diam}(Q)\)  for a cube \(Q\subset {\mathbb {R}}^n\) , then the
desired \(A_1\)  property (REF ) for \(w=\operatorname{dist}(\cdot ,E)^{-\alpha }\)  follows easily from (REF )
by considering a
ball \(B=B(x,r)\)  such that \(x\in E\) ,
\(Q\subset B\)  and \(\vert B\vert \le C(n)\vert Q\vert \) .
On the other hand, if \(\operatorname{dist}(Q,E)\ge 2\operatorname{diam}(Q)\) , then
an argument similar to the one leading to (REF ) shows that (REF ) holds,
and thus \(\operatorname{dist}(\cdot ,E)^{-\alpha }\in A_1\) .
Let \(\lambda >0\)  with \(\operatorname{{Mu}}(E)>\lambda >\alpha \) ,
and let \(x\in E\)  and \(r>0\) .
Observe from inequality \(\operatorname{{Mu}}(E)>0\)  that \(0<h_E(B(x,2r))\le r\) .
Hence, there is \(j_0\in {\mathbb {N}}\) 
such that
\(2^{-j_0} r < h_E(B(x,2r)) \le 2^{1-j_0} r.\) 
Define
\(F_j=\lbrace y\in B(x,r) : \operatorname{dist}(y,E) \le 2^{1-j}r \rbrace  \quad \text{ and }\quad A_{j}=F_{j}\setminus F_{j+1},\) 
for \(j\ge j_0\) .
Since \(\lambda <\operatorname{{Mu}}(E)\) , there is a constant \(C_1=C_1(E,\lambda ,n)\)  such that
\(\begin{split}\frac{\vert F_{j}\vert }{\vert B(x,r)\vert }&\le \frac{2^n\vert E_{2^{2-j}r}\cap B(x,2r)\vert }{\vert B(x,2r)\vert }\\&\le C_1 \biggl (\frac{h_E(B(x,2r))}{2^{-j} r}\biggr )^{-\lambda }= C_1 2^{-j\lambda }\biggl (\frac{h_E(B(x,2r))}{r}\biggr )^{-\lambda }.\end{split}\) 
Since \(\lambda >0\)  and \({\mathchoice{\unknown. {\displaystyle E}\hspace{0.0pt}\vrule height1.593pt}{\hspace{0.0pt}}{0}{.}0\overline{\phantom{\displaystyle E}\displaystyle E}\hspace{0.0pt}}\) E\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)EB(x,r)Fj\( for every \) jj0\(,by letting \) j\( we see in particular that \)\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)EB(x,r) = 0\(.Here \) r>0\( is arbitrary, and thus \)\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E=0\(.\) If \(y\in B(x,r)\setminus {\mathchoice{\unknown. {\displaystyle E}\hspace{0.0pt}\vrule height1.593pt}{\hspace{0.0pt}}{0}{.}0\overline{\phantom{\displaystyle E}\displaystyle E}\hspace{0.0pt}}\) E\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E\(, then\) dist(y,E)y-x<r\(. Hence,\)\(B(y,\operatorname{dist}(y,E))\subset B(x,2r)\setminus E,\)\(and therefore \) 0<dist(y,E)hE(B(x,2r))21-j0r\(.It follows thatthe union of sets \) Aj\( with \) jj0\( cover \) B(x,r)\( up to the set\)\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)EB(x,r)\(, which has measure zero.If \) yAj\(, then\) 2-jr < dist(y,E) 21-jr\(.In addition, \) AjFj\( for every \) jj0\(.By combining the above observations and using~(\ref {e.level_est}) we obtain\begin{equation*}\begin{split}\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}B(x,r)}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{B(x,r)}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{B(x,r)}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{B(x,r)}} \operatorname{dist}(y,E)^{-\alpha }\,dy&\le \frac{1}{\vert B(x,r)\vert } \sum _{j=j_0}^{\infty }\int _{A_{j}}\operatorname{dist}(y,E)^{-\alpha }\,dy \le \sum _{j=j_0}^{\infty }\frac{\vert F_{j}\vert }{\vert B(x,r)\vert }(2^{-j}r)^{-\alpha }\\&\le C_1 \sum _{j=j_0}^{\infty }(2^{-j}r)^{-\alpha } 2^{-j\lambda }\biggl (\frac{h_E(B(x,2r))}{r}\biggr )^{-\lambda }\\&\le C_1 r^{-\alpha }\biggl (\frac{h_E(B(x,2r))}{r}\biggr )^{-\lambda }\sum _{j=j_0}^{\infty }(2^{-j})^{\lambda -\alpha }\\&\le C(C_1,\lambda ,\alpha ) r^{-\alpha }\biggl (\frac{h_E(B(x,2r))}{r}\biggr )^{-\lambda }\left(\frac{h_E(B(x,2r))}{r}\right)^{\lambda -\alpha }\\&\le C(C_1,\lambda ,\alpha ) h_E(B(x,2r))^{-\alpha }\\&\le C(C_1,\lambda ,\alpha ) \operatornamewithlimits{ess\, inf}_{y\in B(x,r)}\operatorname{dist}(y,E)^{-\alpha }.\end{split}\end{equation*}This shows that (\ref {e.suff}) holds, and the claim follows.\)
Theorem 6.6 
Let \(\alpha >0\)  and let \(E\subset {\mathbb {R}}^n\)  be a nonempty
set. Then \(\operatorname{dist}(\cdot ,E)^{-\alpha }\in A_1\)  if and only if \(\operatorname{{Mu}}(E)>\alpha \) .

If \(\operatorname{{Mu}}(E)>\alpha \) , then \(\operatorname{dist}(\cdot ,E)^{-\alpha }\in A_1\)  by Theorem REF .
Conversely, assume that \(\operatorname{dist}(\cdot ,E)^{-\alpha }\in A_1\) .
Then, by the self-improvement of \(A_1\)  weights (see [1]}),
there exists \(s>1\)  such that \(\operatorname{dist}(\cdot ,E)^{-s\alpha }\in A_1\) .
Lemma REF  implies that \(\operatorname{{Mu}}(E)\ge s\alpha >\alpha \) .
Since \(\operatorname{dist}(\cdot ,E)^{0} \in A_1\)  holds for all (nonempty) sets \(E\subset {\mathbb {R}}^n\) 
(under the interpretation that \(0^0=1\) ), Theorem REF 
implies that
\(\operatorname{{Mu}}(E)=\sup \lbrace  \alpha \ge 0 : \operatorname{dist}(\cdot ,E)^{-\alpha } \in A_1 \rbrace \) 
for all nonempty sets \(E\subset {\mathbb {R}}^n\) . On the other hand,
by Theorem REF  we have \(\operatorname{dist}(\cdot ,E)^{-\alpha }\in A_1\) , for some \(\alpha >0\) , if and only if
\(E\)  is weakly porous. This, together with Theorem REF , gives the following corollary.
Corollary 6.7 
A nonempty set \(E\subset {\mathbb {R}}^n\)  is weakly porous
if and only if \(\operatorname{{Mu}}(E)>0\) .

We are now ready for the proof of our second main result.
Since \(E\)  is weakly porous,
we have \(\operatorname{{Mu}}(E)>0\)  by Corollary REF .
Therefore, the equivalences in both (i) and (ii) hold if \(\alpha =0,\) 
and so we may assume from now on that \(\alpha \ne 0.\)
If \(w =\operatorname{dist}(\cdot ,E)^{-\alpha }\in A_1\) , then \(\alpha \ge 0\)  by Lemma REF .
Thus it suffices to consider the case \(\alpha >0\) , and then the claim in (i) follows
directly from Theorem REF .
In part (ii), let \(1<p<\infty \)  and
assume first that \(w\in A_p\) . Because \(E\)  is weakly porous, Lemma REF 
provides us with some \(\sigma >0\)  for which \(\operatorname{dist}(\cdot ,E)^{-\sigma } \in A_1({\mathbb {R}}^n)\) . If \(\alpha > 0,\)  we can use
Lemma REF  with \(\beta =\sigma /\alpha \)  to deduce that \(w=\operatorname{dist}(\cdot , E)^{-\alpha }\in A_1.\) 
Then Theorem REF  implies \(\operatorname{{Mu}}(E) >\alpha ,\)  and so (REF ) holds.
On the other hand, if \(\alpha <0,\)  then
we have
\(\operatorname{dist}(\cdot ,E)^{- \left(\frac{-\alpha }{p-1}\right) }= w^{1-p^{\prime }} \in A_{p^{\prime }},\) 
where \(\frac{-\alpha }{p-1} >0.\) 
Hence the previous case, for a positive power and the class \(A_{p^{\prime }}\) ,
shows that
\((1-p^{\prime })\operatorname{{Mu}}(E) < 0 < \frac{-\alpha }{p-1} < \operatorname{{Mu}}(E),\) 
which is equivalent to (REF ).
Conversely, assume that (REF ) holds for some \(\alpha \ne 0.\)  If \(\alpha > 0,\) 
then \(w=\operatorname{dist}(\cdot ,E)^{-\alpha } \in A_1\subset A_p\)  by Theorem REF .
Finally, if \(\alpha <0,\)  we observe that (REF ) is equivalent to (REF ),
where \(\frac{-\alpha }{p-1} >0\) . Thus we may apply the preceding case for the exponent \(\frac{-\alpha }{p-1}>0\)  and the class \(A_{p^{\prime }}\)  to conclude that \(\operatorname{dist}(\cdot , E)^{\alpha /(p-1)} \in A_{p^{\prime }}.\)  Hence \(w = \operatorname{dist}(\cdot ,E)^{-\alpha } \in A_p,\) 
proving part (ii).
Remark 6.8 
Note that in part (i) of Theorem REF  the explicit assumption that
\(E\)  is weakly porous is needed in the necessity part, since for \(\alpha =0\) 
the claim \(w\in A_1\)  holds for all (nonempty) sets \(E\subset {\mathbb {R}}^n\) . However, if \(\alpha >0\) ,
then we know by Theorem REF 
that \(w\in A_1\)  can only hold if \(E\)  is weakly porous,
which in turn is equivalent to \(\operatorname{{Mu}}(E)>0\) .
In part (ii) the case \(\alpha =0\)  again shows that (REF ) is
not necessary for \(w\in A_p\) , for general sets \(E\subset {\mathbb {R}}^n\) .
Moreover, if we do not assume weak porosity of \(E\) , then even
in the case \(\alpha \ne 0\)  the requirement (REF )
is not necessary for \(w\in A_p\) . This follows from Theorem REF ,
which gives
a set \(E\subset {\mathbb {R}}\)  with \(\operatorname{{Mu}}(E)=0\) ,
i.e. \(E\)  is not weakly porous, such that
\(\operatorname{dist}(\cdot ,E)^{-\alpha }\in A_p\)  for all \(0<\alpha <1\)  and all \(1<p<\infty \) .

Remark 6.9 
When \(E\subset {\mathbb {R}}^n\)  is a bounded set, the
upper Minkowski (or box) dimension
\(\operatorname{\overline{dim}_M}(E)\) 
is the infimum of all \(\lambda \ge 0\)  for which there is a constant \(C\)  such that
\(N(E,r)\le Cr^{-\lambda }\) 
for every \(0<r<\operatorname{diam}(E)\) .
Note that (REF )
is equivalent to the condition that there is a constant \(C\)  such that
\(\vert E_r\vert \le Cr^{n-\lambda }\) 
for every \(0<r<\operatorname{diam}(E)\) ; this follows from Lemma REF .
If a set \(E\subset {\mathbb {R}}^n\)  is weakly porous and \(0<\alpha <\operatorname{{Mu}}(E)\) ,
then \(\operatorname{dist}(\cdot ,E)^{-\alpha }\in A_1\)  by Theorem REF ,
and so \(\int _{B(x,R)} \operatorname{dist}(y,E)^{-\alpha }\,dy<\infty \) 
for every \(x\in E\)  and \(R>0\) . Hence, if \(x\in E\)  and \(R>0\) , then
it holds for all \(0<r<\operatorname{diam}(E\cap B(x,R))\le 2R\)  that
\(\vert (E\cap B(x,R))_r\vert \le r^{\alpha }\int _{B(x,3R)} \operatorname{dist}(y,E)^{-\alpha }\,dy\le C(x,R,E)r^{n-(n-\alpha )}.\) 

Thus
\(\operatorname{\overline{dim}_M}(E\cap B(x,R))\le n-\alpha <n.\) 
Since this holds for all \(0<\alpha <\operatorname{{Mu}}(E)\) ,
we obtain \(\operatorname{\overline{dim}_M}(E\cap B(x,R))\le n-\operatorname{{Mu}}(E)\) .
In particular, if \(E\subset {\mathbb {R}}^n\)  is bounded, then
\(0\le \operatorname{{Mu}}(E)\le n - \operatorname{\overline{dim}_M}(E)\) .
On the other hand, the condition that \(\operatorname{\overline{dim}_M}(E\cap B(x,R))\le c < n\)  for every \(x\in E\)  and \(R>0\)  is not sufficient
for the weak porosity of \(E\) . For instance, if \(E\subset {\mathbb {Z}}\subset {\mathbb {R}}\)  is not weakly porous
(e.g. \(E={\mathbb {N}}\) ), then we have \(\operatorname{\overline{dim}_M}(E\cap B(x,R)) = 0 < 1 = n\) 
for every \(x\in E\)  and \(R>0\)  since \(E\cap B(x,R)\)  is a finite set.
See also [2]} and the references therein for much more elaborate
connections between Minkowski dimensions and the integrability of distance functions.

Example of a weakly porous set
The notions of weak porosity and Muckenhoupt exponent are interesting only if there are
(plenty of) weakly porous sets which are not porous. Below we construct a family of such sets in \({\mathbb {R}}^n\) 
and determine the Muckenhoupt exponents for different values of the parameter \(\gamma >0\) .
These sets are inspired by the often used one-dimensional
example \(\lbrace j^{-\gamma }:j\in {\mathbb {N}}\rbrace \cup \lbrace 0 \rbrace \subset {\mathbb {R}}\) .
Theorem 7.1 
Let \(n\in {\mathbb {N}}\)  and \(\gamma >0\) . Then
the set
\(E=\bigcup _{j = 1} ^\infty \partial B\big (0, j^{-\gamma } \big ) \cup \lbrace 0 \rbrace \subset {\mathbb {R}}^n\) 
is weakly porous with \(\operatorname{{Mu}}(E)=\min \lbrace 1, \frac{n \gamma }{1+\gamma } \rbrace .\)

The origin is included in \(E\)  in order to have a compact set,
but for our purposes this does not make any essential difference.
See Figure REF  for an illustration of
the set
\(E\) .
<FIGURE>By considering the balls \(B(0,j^{-\gamma })\)  as \(j\rightarrow \infty \) , it is
straightforward to verify that \(E\)  is not porous,
and hence \(\operatorname{dim_A}(E)=n\) . Moreover,
special cases of the
computations in the proof of
Theorem REF  below
can be used to show that \(\operatorname{\overline{dim}_M}(E)=\max \lbrace n-1,\frac{n}{1+\gamma }\rbrace \) ,
and so in combination with Theorem REF 
we obtain for the set \(E\)  the identity \(\operatorname{{Mu}}(E) = n - \operatorname{\overline{dim}_M}(E)\) ;
compare to Remark REF .
For the proof of Theorem REF , we
define \(S^t=\partial B(0,t)\)  and \(A_s^t={\mathchoice{\unknown. {\displaystyle B}\hspace{0.0pt}\vrule height1.593pt}{\hspace{0.0pt}}{0}{.}0\overline{\phantom{\displaystyle B}\displaystyle B}\hspace{0.0pt}}\) B\(\) B\(\) 

B
\(\)B\(\) B\(\) 

B
\(\)B\(\) B\(\) 

B
\(\)B ( 0, t ) B(0, s )\( for every \) 0s t\(, where we use the notation \) B(0,0)=\(.We begin with the following lemma.\)
Lemma 7.2 
Let \(B=B(x,R) \subset {\mathbb {R}}^n\)  be a ball such that \(x \in S^t,\)  with \(t=j^{-\gamma }\)  for some \(j\in {\mathbb {N}}\) , and \(B \cap E = B \cap S^t.\)  Then (REF ) holds for \(B\)  if and only if \(\alpha \le 1\) . Moreover, if \(\alpha \le 1\) , then
the constant in (REF ) for \(B\) 
can be chosen to depend on \(n\) , \(\gamma \)  and \(\alpha \)  only.

We have \(h_E(B) = R/2,\)  and given \(0<r<h_E(B),\)  the set \(A_{j^{-\gamma }-r}^{j^{-\gamma }+r} \cap B\)  satisfies
\((2r) \inf _{b\in [t-r,t+r]} \mathcal {H}^{n-1}\left( S^{b} \cap B \right) \le \vert A_{j^{-\gamma }-r}^{j^{-\gamma }+r} \cap B \vert \le (2r) \sup _{b\in [t-r,t+r]} \mathcal {H}^{n-1}\left( S^{b} \cap B \right),\) 
where \(\mathcal {H}^{n-1}\)  is the normalized
Hausdorff measure in \({\mathbb {R}}^n\) .
For each \(b\in [t-r,t+r],\)  the set \(S^b \cap B\)  is a hyperspherical cap within the sphere \(S^{b},\)  whose angle \(\alpha _{b}\)  satisfies, by virtue of the law of cosines, that \(\cos (\alpha _b) = \frac{b^2+t^2-R^2}{2bt}.\)  Therefore
\(\sin \left( \frac{\alpha _b}{2} \right) = \left( \frac{R^2-(b-t)^2}{4bt} \right)^{1/2}.\) 
For a sufficiently small constant \(c(\gamma ),\)  we have that \(r \le c(\gamma ) h_E(B)\)  implies \(\alpha _b \simeq C(\gamma ) \left( \frac{R }{2t} \right)\)  for every \(b\in [t-r,t+r]\) ;
here and below \(a\simeq C(*) b\)  means that \(C(*)^{-1}b\le a\le C(*)b\) . This leads us to
\(\mathcal {H}^{n-1}\big ( S^{b} \cap B \big ) \simeq C(n,\gamma ) b^{n-1} \left( \alpha _b \right)^{n-1} \simeq C(n,\gamma ) b^{n-1} \left( \frac{R}{2t} \right)^{ n-1 } \simeq C(n,\gamma ) R^{n-1},\) 
for every \(b\in [t-r,t+r].\)  The sets \(A_{(j-1)^{-\gamma }-r}^{(j-1)^{-\gamma }} \cap B\)  and \(A_{(j+1)^{-\gamma }}^{(j+1)^{-\gamma }+r} \cap B\)  (meaning \(A_{(j-1)^{-\gamma }-r}^{(j-1)^{-\gamma }} = \emptyset \)  in the case \(j=1\) ) are also contained in \(E_r \cap B\) , but their measures are controlled by \(C(n,\gamma ) \vert A_{j^{-\gamma }-r}^{j^{-\gamma }+r} \cap B \vert .\)  Bearing in mind this observation and (REF ) and (REF ), we obtain
\(\left( \frac{h_E(B)}{r} \right)^{\alpha } \frac{\vert E_r \cap B \vert }{\vert B \vert } \le C(n,\gamma ,\alpha ) R^{\alpha -n} r^{-\alpha } \vert A_{j^{-\gamma }-r}^{j^{-\gamma }+r} \cap B \vert \le C(n,\gamma ,\alpha ) \left( \frac{r}{R} \right)^{1-\alpha }.\) 
If \(\alpha \le 1,\)  the last term is bounded by \(C(n,\gamma ,\alpha ).\)  On the other hand, if \(\alpha > 1,\)  then (REF ) and (REF ) yield
\(\left( \frac{h_E(B)}{r} \right)^{\alpha } \frac{\vert E_r \cap B \vert }{\vert B \vert } \ge c(n,\gamma ,\alpha ) R^{\alpha -n} r^{1-\alpha } R^{n-1} \ge c(n,\gamma ,\alpha ) \left( \frac{r}{R} \right)^{1-\alpha },\) 
and the last term tends to infinity as \(r \rightarrow 0.\)
First we show that (REF ) holds for every \(\alpha \)  with \(0<\alpha <\min \lbrace 1, \frac{n\gamma }{1+\gamma } \rbrace .\) 
This implies that
\(\operatorname{{Mu}}(E) \ge \min \lbrace 1, \frac{n \gamma }{1+\gamma } \rbrace >0\) ,
and thus \(E\)  is weakly porous, by Corollary REF .
Fix \(0<\alpha <\min \lbrace 1, \frac{n\gamma }{1+\gamma } \rbrace \)  and let
\(B=B(x,R) \subset {\mathbb {R}}^n\)  be a ball with \(x\in E,\)  and let \(0<r<h_E(B)\) .
We suppose first that \(B\)  is contained in \({\mathchoice{\unknown. {\displaystyle B}\hspace{0.0pt}\vrule height1.593pt}{\hspace{0.0pt}}{0}{.}0\overline{\phantom{\displaystyle B}\displaystyle B}\hspace{0.0pt}}\) B\(\) B\(\) 

B
\(\)B\(\) B\(\) 

B
\(\)B\(\) B\(\) 

B
\(\)B(0,1).\( Let \) k\( be the largest number in \) N\( and \) N\( be the smallest number in \) N\( such that \) B \(\) B\(\) 

B
\(\)B\(\) B\(\) 

B
\(\)B\(\) B\(\) 

B
\(\)B\(\) B\(\) 

B
\(\)B ( 0, k- ) B(0, N- ).\( We interpret \) N-=0\( and \) B(0, N- ) =\( when \) 0 \(\) B\(\) 

B
\(\)B\(\) B\(\) 

B
\(\)B\(\) B\(\) 

B
\(\)B\(\) B\(\) 

B
\(\)B.\( It is clear that \) N k+2\(, since the center \) x\( of \) B\( belongs to \) E\(. In the case\) N= k+2\( we have \) xS(k+1)-\(,and (\ref {e.muckdim}) follows immediately from Lemma~\ref {lemmaonecircle}.Hence we may assume that \) N k+3.\( Also, observe that\begin{equation}h_E(B) \le \tfrac{1}{2}\left( k^{-\gamma }-(k+1)^{-\gamma } \right) \le \tfrac{\gamma }{2} k^{-\gamma -1}\end{equation}and\begin{equation}R \ge \tfrac{1}{2} \left( (k+1)^{-\gamma }-(N-1)^{-\gamma } \right)\ge \tfrac{\gamma }{2}(N-k-2)(N-1)^{-\gamma -1}.\end{equation}Now we study two cases.\)
(i) Suppose \(\operatorname{dist}(\lbrace 0 \rbrace , B ) > \operatorname{diam}(B).\)  We have the estimates
\((k+1)^{-\gamma } \le \sup _{x\in B} \vert x\vert \le \operatorname{dist}(\lbrace 0 \rbrace , B ) + \operatorname{diam}(B)\le 2 \operatorname{dist}(\lbrace 0 \rbrace , B ) \le 2 (N-1)^{-\gamma } ,\) 
and so \(N-1\le C(\gamma ) (k+1).\)  Then we have
\(\vert E_r \cap B \vert & \le \sum _{j=k}^N \vert A_{j^{-\gamma }-r}^{j^{-\gamma }+r} \cap B \vert \le C(n) \sum _{j=k}^N r R^{n-1} \le C(n) (N-k+1) r R^{n-1}.\) 

The previous observation, together with
() and (), leads us to
\(\left( \frac{h_E(B)}{r} \right)^{\alpha } \frac{\vert E_r \cap B \vert }{\vert B \vert } & \le C(n,\gamma ) k^{-(1+\gamma ) \alpha } \, r^{1-\alpha } R^{-1}(N-k+1) \\&\le C(n,\gamma ) k^{-(1+\gamma ) \alpha } \, r^{1-\alpha }(N-1)^{1+\gamma }\\&\le C(n,\gamma ) k^{-(1+\gamma ) \alpha } \, r^{1-\alpha }(k+1)^{1+\gamma }\\& \le C(n,\gamma ) \left(rk^{1+\gamma }\right)^{1-\alpha }.\) 

The last term is bounded by a constant \(C(n, \gamma ,\alpha )\)  because \(\alpha \le 1\)  and \(r \le C(\gamma ) k^{-1-\gamma }.\)
(ii) Now suppose \(\operatorname{dist}(\lbrace 0 \rbrace , B ) \le \operatorname{diam}(B).\)  Then we have
\((2k)^{-\gamma } \le (k+1)^{-\gamma } \le \operatorname{dist}(\lbrace 0 \rbrace , B ) + \operatorname{diam}(B) \le 2 \operatorname{diam}(B),\) 
and hence \(k^{-\gamma } \le 2^{1+\gamma } \operatorname{diam}(B).\)  Given \(0<r<h_E(B),\)  denote by \(j_0\in {\mathbb {N}}\)  the
smallest number for which
\(2r \ge j_0^{-\gamma }-\left( j_0+1 \right)^{-\gamma } \ge C(\gamma )(j_0+1)^{-\gamma -1}.\) 
Notice that \(k< j_0\)  and, by the definition of \(j_0,\)  we also have
\(r \le (j_0-1)^{-\gamma }- j_0^{-\gamma }\le C(\gamma ) (j_0-1)^{-\gamma -1}\le C(\gamma )j_0^{-\gamma -1}.\) 
This observation permits us to write
\(\vert E_r \cap B \vert & \le \vert B \cap A_{k^{-\gamma } -r}^{k^{-\gamma }} \vert + \vert B(0,j_0^{-\gamma }+r) \cap B \vert + \sum _{j=k+1}^{j_0-1} \vert A_{j^{-\gamma }-r}^{j^{-\gamma }+r} \cap B \vert \\& \le C(n) \biggl ( \left( j_0^{-\gamma }+r \right)^n + \sum _{j=k }^{j_0-1} r \left( j^{-\gamma }+r \right)^{n-1} \biggr ) .\) 

Using the inequalities \(0<\alpha <\min \lbrace 1, \frac{n\gamma }{1+\gamma } \rbrace ,\)  \(k^{-\gamma } \le C(\gamma ) R\) , \(c(\gamma )j_0^{-1-\gamma }\le r \le C(\gamma )j_0^{-1-\gamma },\)  and \( h_E(B) \le C(\gamma ) k^{-1-\gamma },\)  we obtain
\(\left( \frac{h_E(B)}{r} \right)^{\alpha } \frac{\vert E_r \cap B \vert }{\vert B \vert } & \le C(n,\gamma ) k^{n \gamma -(1+\gamma )\alpha } r^{-\alpha }\biggl ( \left( j_0^{-\gamma }+r \right)^n + \sum _{j=k }^{j_0-1} r \left( j^{-\gamma }+r \right)^{n-1} \biggr ) \\& \le C(n,\gamma ) k^{n \gamma -(1+\gamma )\alpha } r^{-\alpha } \biggl ( j_0^{-n\gamma } + \sum _{j=k }^{j_0-1} r \left( j^{-\gamma }+r \right)^{n-1} \biggr ) \\& \le C(n,\gamma ) \biggl ( \left( k j_0^{-1} \right)^{n\gamma -(1+\gamma ) \alpha } + k^{n \gamma -(1+\gamma )\alpha } \sum _{j=k }^{j_0-1} r^{1-\alpha } \left( j^{-\gamma }+r \right)^{n-1} \biggr ) \\& \le C(n,\gamma )+ C(n,\gamma ) k^{n \gamma -(1+\gamma )\alpha } \sum _{j=k }^{j_0-1} j^{-(1-\alpha )(1+\gamma )} \left( j^{-\gamma }+j^{-1-\gamma }\right)^{n-1} \\& \le C(n,\gamma )+ C(n,\gamma ) k^{n \gamma -(1+\gamma )\alpha } \sum _{j=k }^{\infty } j^{-1-n\gamma +(1+\gamma ) \alpha } \le C(n,\gamma ,\alpha ),\) 

where the last inequality follows
by comparing the series to
\(\int _{k}^\infty t^{-1-n\gamma +(1+\gamma ) \alpha }\,dt\) , bearing in mind that \(\alpha < \frac{n\gamma }{1+\gamma }.\) 
The cases (i) and (ii) together show that
(REF ) holds when \(B\subset {\mathchoice{\unknown. {\displaystyle B}\hspace{0.0pt}\vrule height1.593pt}{\hspace{0.0pt}}{0}{.}0\overline{\phantom{\displaystyle B}\displaystyle B}\hspace{0.0pt}}\) B\(\) B\(\) 

B
\(\)B\(\) B\(\) 

B
\(\)B\(\) B\(\) 

B
\(\)B(0,1)\(.\)
Now suppose that \(B=B(x,R)\)  is not contained in \({\mathchoice{\unknown. {\displaystyle B}\hspace{0.0pt}\vrule height1.593pt}{\hspace{0.0pt}}{0}{.}0\overline{\phantom{\displaystyle B}\displaystyle B}\hspace{0.0pt}}\) B\(\) B\(\) 

B
\(\)B\(\) B\(\) 

B
\(\)B\(\) B\(\) 

B
\(\)B(0,1).\(In the case \) r1-2-2\( we use the fact that \) n-> 0\( to estimate{\begin{@align*}{1}{-1}\left( \frac{h_E(B)}{r} \right)^{\alpha } \frac{\vert E_r \cap B \vert }{\vert B \vert }& \le C(n)\vert E_r \vert r^{-\alpha } R^{\alpha -n}\le C(n)\vert {\mathchoice{\unknown. {\displaystyle B}\hspace{0.0pt}\vrule height1.593pt}{\hspace{0.0pt}}{0}{.}0\overline{\phantom{\displaystyle B}\displaystyle B}\hspace{0.0pt}}\end{@align*}{B}}{\unknown. {\textstyle B}\hspace{0.0pt}\vrule height1.593pt}\hspace{0.0pt}\overline{\phantom{\textstyle B}\textstyle B}\hspace{0.0pt}\)\(\) B\(\) B\(\) 

B
\(\)B\(\) B\(\) 

B
\(\)B(0,r+1)r- R-n
C(n,)(rR)n-C(n,,).

In the sequel, we will assume that
\(r< \frac{1-2^{-\gamma }}{2}\) .
If \(x\in E \setminus S^1,\)  then \(R\ge h_E(B) \ge c(\gamma ) R \ge c(\gamma )\) 
and
\(\vert E_r \cap B \vert \le \vert E_r \cap B(0,1)\vert + \vert E_r \setminus B(0,1)\vert \le \vert E_r \cap B(0,1)\vert +C(n)r.\) 
Therefore
\(\left( \frac{h_E(B)}{r} \right)^{\alpha } \frac{\vert E_r \cap B \vert }{\vert B \vert }\le C(n)(r+\vert E_r \cap B(0,1)\vert ) r^{-\alpha } R^{\alpha -n}\le C(n,\gamma ,\alpha ) R^{\alpha -n} \le C(n,\gamma ,\alpha ),\) 
where the second inequality follows
by using the above case (ii) with \(B=B(0,1)\) .
If \(x\in S^1\)  and \(R\ge \frac{1-2^{-\gamma }}{2},\)  then we can repeat the preceding argument to show that (REF ) holds, and finally, if \(x\in S^1\)  and \(R<\frac{1-2^{-\gamma }}{2},\) 
then (REF ) holds by Lemma REF .
Next we show that \(\operatorname{{Mu}}(E) \le \min \lbrace 1, \frac{n \gamma }{1+\gamma } \rbrace .\) 
The bound \(\operatorname{{Mu}}(E) \le 1\)  follows from Lemma REF .
Let \(\alpha > \frac{n \gamma }{1+\gamma }\)  and consider the ball \(B=B(0,1).\) 
Then \(h_E(B)=\frac{1-2^{-\gamma }}{2}=C(\gamma ).\)  Given \(0<r<\frac{1}{100}\) , let \(j_0\in {\mathbb {N}}\)  be the
smallest number for which \(2r \ge j_0^{-\gamma }-(j_0+1)^{-\gamma }.\)  Then \(r\)  is comparable to \(c(\gamma )j_0^{-1-\gamma }\)  and the annuli \(\lbrace A_{j^{-\gamma }-r}^{j^{-\gamma }+r} \rbrace _{j=1}^{j_0}\)  are pairwise disjoint.
For sufficiently small \(r,\)  we thus have
\(\left( \frac{h_E(B)}{r} \right)^{\alpha } \frac{\vert E_r \cap B \vert }{\vert B \vert } &\ge c(n,\gamma ,\alpha ) r^{-\alpha }\sum _{j=2}^{j_0-1} \left( \left( j^{-\gamma }+r \right)^{n}- \left( j^{-\gamma }-r \right)^{n} \right) \\& \ge c(n,\gamma ,\alpha ) r^{1-\alpha } \sum _{j=2}^{j_0-1} \left( j^{-\gamma }-r \right)^{n-1} \\& \ge c(n,\gamma ,\alpha ) r^{1-\alpha } j_0 \left( (j_0-1)^{-\gamma }-r \right)^{n-1} \\& \ge c(n,\gamma ,\alpha ) r^{1-\alpha } j_0 \left( j_0^{-\gamma }-j_0^{-\gamma -1} \right)^{n-1} \ge c(n,\gamma ,\alpha ) r^{1-\alpha } j_0^{1-\gamma (n-1)} \\&\ge c(n,\gamma ,\alpha ) j_0^{(1-\alpha )(-1-\gamma )} j_0^{1-\gamma (n-1)} = c(n,\gamma ,\alpha ) j_0^{(1+\gamma ) \alpha -n \gamma }.\) 

The last term goes to infinity as \(r\rightarrow 0,\)  since \(\alpha > \frac{n \gamma }{1+\gamma }\) .
Hence (REF ) does not
hold if \(\alpha > \frac{n \gamma }{1+\gamma }\) , showing
that \(\operatorname{{Mu}}(E) \le \frac{n \gamma }{1+\gamma }\) .

\(A_p\) -distance set that is not weakly porous
In this section we construct a set \(E \subset {\mathbb {R}}\)  such
that \(\operatorname{dist}(\cdot ,E)^{-\alpha } \in A_p\setminus A_1\) 
for all \(0<\alpha <1\)  and all \(1<p<\infty \) ; see
Theorem REF .
Recall that we abbreviate \(d_E=\operatorname{dist}(\cdot ,E)\) .
Let \(E_0=\lbrace 0,1 \rbrace \)  and write \(t_n = 1-\frac{1}{2n}\)  for every \(n \in {\mathbb {N}}\) .
Then, for every \(n\in {\mathbb {N}},\) 
the set \(E_n\)  is defined as \(E_n=E_{n-1} \cup E_{n-1}^1 \cup E_{n-1}^2,\)  where:

\(E_{n-1}^1\)  is a translation of \(E_{n-1}\)  dilated by the factor \(t_n\)  and whose first point is the last point of \(E_{n-1},\)

\(E_{n-1}^2\)  is a translation of \(E_{n-1}\)  whose first point is the last point of \(E_{n-1}^1.\)

Finally, we define \(E^+= \bigcup _{n=0}^\infty E_n\)  and \(E= E^+ \cup \left( - E^+ \right).\)  Here \(-E^+\)  is the reflection of \(E^+\)  with respect to the origin. We let \(Q_n,\)  \(Q_n^1,\)  and \(Q_n^2\)  denote the smallest intervals containing \(E_n\) , \(E_n^1,\)  and \(E_n^2\)  respectively, for every \(n\in {\mathbb {N}}\cup \lbrace 0 \rbrace .\) 
See Figure REF  for an illustration of the first steps of the construction.
<FIGURE>During the rest of this section, we prove the following theorem
for the set \(E\) .
Theorem 8.1 
Let \(E\subset {\mathbb {R}}\)  be as constructed above.
Then it holds for all \(0<\alpha <1\)  and all \(1<p<\infty \) 
that \(\operatorname{dist}(\cdot ,E)^{-\alpha } \in A_p \setminus A_1\) .
In particular,
the set \(E\)  is not weakly porous and \(\operatorname{{Mu}}(E)=0\) .

Let \(0<\alpha <1\)  and \(1<p<\infty \) . We show in Lemma REF  that
\(\operatorname{dist}(\cdot ,E)^{-\alpha } \notin A_1\) , and the claim
\(\operatorname{dist}(\cdot ,E)^{-\alpha } \in A_p\) 
follows from Lemma REF .
Since \(\operatorname{dist}(\cdot ,E)^{-\alpha } \notin A_1\)  for every \(\alpha >0\) , the set
\(E\)  is not weakly porous by Theorem REF ,
and thus Corollary REF  implies
that \(\operatorname{{Mu}}(E)=0\) .
We say that a closed interval \(I\)  is an edge of \(E\) 
if the endpoints of \(I\)  are two consecutive points of \(E.\) 
For every \(n\in {\mathbb {N}}\cup \lbrace 0 \rbrace ,\)  the following
properties hold:

Each of the cubes \(Q_n,\)  \(Q_n^1,\)  and \(Q_n^2\)  has \(3^n\)  edges of \(E,\)  of which the middle ones for \(n\ge 1\)  have lengths equal to \(t_1 t_2 \cdots t_n,\)  \(t_1 t_2 \cdots t_n t_{n+1},\)  and \(t_1 t_2 \cdots t_n\) , respectively.

Each of the cubes \(Q_n\)  and \(Q_n^2\)  contains translated copies of the intervals \(Q_0, \ldots , Q_{n-1}\)  distributed in a palindromic manner: both \(Q_n\)  and \(Q_n^2\)  contain from left to right as well as from right to left intervals \(Q_0^* \subset Q_1^* \subset \cdots \subset Q_{n-1}^*\)  that are translated copies of \(Q_0 \subset Q_1 \subset \cdots \subset Q_{n-1}\) , respectively.

Each cube \(Q_n^1\)  contains from left to right as well as from right to left intervals \(t_N Q_0^* \subset t_N Q_1^* \subset \cdots \subset t_N Q_{n-1}^*\)  that are translated copies of \(Q_0 \subset Q_1 \subset \cdots \subset Q_{n-1}\)  dilated by \(t_N.\)

\(d_E= d_{E_n}\)  on \(Q_n.\)

\(\vert Q_n \vert = (2+t_n) \vert Q_{n-1} \vert \)  for every \(n\in {\mathbb {N}}.\)

Lemma 8.2 
For every \(n \in {\mathbb {N}}\)  and every \(\beta >-1,\)  we have
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_n}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_n}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_n}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_n}} d_E(x)^\beta \,dx = \frac{2+t_n^{1+\beta }}{2+t_n} \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_{n-1}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_{n-1}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_{n-1}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_{n-1}}} d_E(x)^\beta \,dx.\) 

Let \(n\in {\mathbb {N}}\)  and \(\beta >-1\) .
By the construction of \(E\)  and the definition of \(Q_n\) , we obtain
\(\int _{Q_n} d_E^\beta & =\int _{Q_n} d_{E_n}^\beta = \int _{Q_{n-1}} d_{E_{n-1}}^\beta + \int _{Q_{n-1}^1} d_{E_{n-1}^1}^\beta + \int _{Q_{n-1}^2} d_{E_{n-1}^2}^\beta \\& = \left( 2 + t_n^{1+\beta } \right) \int _{Q_{n-1}} d_{E_{n-1}}^\beta = \left( 2 + t_n^{1+\beta } \right) \int _{Q_{n-1}} d_{E}^\beta .\) 

The claim follows by combining the above identity with the fact \(\vert Q_n \vert = (2+t_n) \vert Q_{n-1} \vert \) .
Lemma 8.3 
For every \(0<\alpha <1\)  and \(1<p<\infty \) , there exists \(N_0\in {\mathbb {N}}\) , only depending on \(\alpha \)  and \(p\) , for which
\(\log \left( \frac{ 2+t_n^{1-\alpha }}{2+t_n} \right) \ge \frac{\alpha }{ 12n} \quad \text{and} \quad \log \left[\left( \frac{ 2+t_n^{1-\alpha }}{2+t_n} \right) \left( \frac{2+t_n^{1+\frac{\alpha }{p-1}}}{2+t_n} \right)^{p-1}\right] \le \frac{\alpha ^2 p}{18(p-1) n^2}\) 
for every \(n \ge N_0.\)

Consider the functions
\(f(t)=\log \left( \frac{ 2+t^{1-\alpha }}{2+t} \right),\qquad g(t)=\log \left[\left( \frac{ 2+t^{1-\alpha }}{2+t} \right) \left( \frac{2+t^{1+\frac{\alpha }{p-1}}}{2+t} \right)^{p-1}\right]\) 
for \(t>0.\)  These functions satisfy \(f(1)=0,\)  \(f^{\prime }(1)= -\frac{\alpha }{3},\)  \(g(1)=g^{\prime }(1)=0\)  and \(g^{\prime \prime }(1)= \frac{2\alpha ^2 p}{9(p-1)}.\)  Let \(\varepsilon \in (0,1/2)\)  be small enough so that \(\vert t-1 \vert \le \varepsilon \)  implies
\(\vert f(t)-f(1)-f^{\prime }(1)(t-1) \vert \le \frac{\alpha }{6} \vert t-1 \vert \) 
and
\(\vert g(t)-g(1)-g^{\prime }(1)(t-1)-\tfrac{1}{2}g^{\prime \prime }(1)(t-1)^2 \vert \le \frac{\alpha ^2 p}{9(p-1)} \vert t-1 \vert ^2.\) 
Taking \(N_0 \in {\mathbb {N}}\)  large enough so that \(N_0 \ge 1/(2\varepsilon )\) 
it follows that \(\vert 1-t_n\vert \le \varepsilon \)  for every \(n \ge N_0,\) 
and so the above estimates yield
\(f(t_n) \ge \frac{\alpha }{12n} \qquad \text{and} \qquad g(t_n)\le \frac{\alpha ^2 p}{18(p-1) n^2}. \) 
Lemma 8.4 
For every \(0<\alpha <1\) , the weight \(d_E^{-\alpha }\)  does not belong to \(A_1.\)

Let \(N_0\)  be the constant in Lemma REF 
with, say, \(p=2\) ; the value of \(p\)  is irrelevant here. Applying repeatedly
Lemma REF , we obtain, for every \(n\in {\mathbb {N}},\)
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_n}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_n}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_n}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_n}} d_E^{-\alpha } = \left( \prod _{k=1}^n \frac{2+t_k^{1-\alpha }}{2+t_k} \right) \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}} d_E^{-\alpha } \ge \left( \prod _{k=N_0}^n \frac{2+t_k^{1-\alpha }}{2+t_k} \right) \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}} d_E^{-\alpha }.\) 
By the first inequality of Lemma REF , we have
\(\log \left( \prod _{k=N_0}^n \frac{2+t_k^{1-\alpha }}{2+t_k} \right) = \sum _{k=N_0}^n \log \left( \frac{2+t_k^{1-\alpha }}{2+t_k} \right) \ge \sum _{k=N_0}^n \frac{\alpha }{ 12 k},\) 
and it follows that
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_n}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_n}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_n}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_n}} d_E^{-\alpha }\ge \exp \left(\sum _{k=N_0}^n \frac{\alpha }{ 12 k}\right)\,\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}} d_E^{-\alpha }.\) 
Since the harmonic series
diverges, we see that
\(\lim _{n\rightarrow \infty } \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_n}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_n}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_n}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_n}} d_E^{-\alpha } =\infty \) . On the other hand, each \(Q_n\)  contains edges of \(E\)  of length equal to 1, and thus \(\operatornamewithlimits{ess\, inf}_{Q_n} d_E^{-\alpha } =2^\alpha .\)  We conclude that \(d_E^{-\alpha } \notin A_1.\)
Lemma 8.5 
For every \(0<\alpha <1\)  and \(1<p<\infty \) ,
there exists a constant \(\widehat{C} = \widehat{C}(\alpha ,p)>0\)  such that
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}} d_E(x)^{-\alpha }\,dx \left( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}} d_E(x)^{\frac{\alpha }{p-1}}\,dx \right)^{p-1} \le \widehat{C}\) 
for every \(N\in {\mathbb {N}}\cup \lbrace 0 \rbrace \) .

For \(N=0\)  the claim is clear. Assume that \(N\ge 1\) . By Lemma REF ,
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}} d_E^{-\alpha }\left( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}} d_E^{\frac{\alpha }{p-1}} \right)^{p-1} & = \left( \prod _{n=1}^{N} \frac{2+t_n^{1-\alpha }}{2+t_n} \right) \left( \prod _{n=1}^{N} \frac{2+t_n^{1+\frac{\alpha }{p-1}}}{2+t_n} \right)^{p-1} \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}} d_E^{-\alpha }\left( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}} d_E^{\frac{\alpha }{p-1}} \right)^{p-1} \\& =\prod _{n=1}^{N} \left( \frac{2+t_n^{1-\alpha }}{2+t_n} \right) \left( \frac{2+t_n^{1+\frac{\alpha }{p-1}}}{2+t_n} \right)^{p-1}\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}} d_E^{-\alpha }\left( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}} d_E^{\frac{\alpha }{p-1}} \right)^{p-1}.\) 

Let \(N_0=N_0(\alpha ,p)\in {\mathbb {N}}\)  be as in Lemma REF .
Then
\(\log & \prod _{n=1}^{N} \left( \frac{2+t_n^{1-\alpha }}{2+t_n} \right) \left( \frac{2+t_n^{1+\frac{\alpha }{p-1}}}{2+t_n} \right)^{p-1} \\& \le \sum _{n=1}^{N_0-1} \log \left[\left( \frac{2+t_n^{1-\alpha }}{2+t_n} \right) \left( \frac{2+t_n^{1+\frac{\alpha }{p-1}}}{2+t_n} \right)^{p-1}\right]+ \sum _{n=N_0}^N \frac{\alpha ^2 p}{18(p-1) n^2},\) 

where the right-hand side is bounded from above by a constant \(C_1 = C_1(\alpha ,p)\)  independent of \(N\) .
Hence,
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}} d_E^{-\alpha }\left( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}} d_E^{\frac{\alpha }{p-1}} \right)^{p-1} \le e^{C_1} \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}} d_E^{-\alpha }\left( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}} d_E^{\frac{\alpha }{p-1}} \right)^{p-1},\) 
and the claim follows.
Lemma 8.6 
For every \(0<\alpha <1\)  and \(1<p<\infty \) ,
there exists a constant \(C = C(\alpha ,p)>0\)  such that
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }} d_E(x)^{-\alpha }\,dx \left( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }} d_E(x)^{\frac{\alpha }{p-1}}\,dx \right)^{p-1} \le C\) 
for every cube \(Q \subset [0, +\infty ).\)

Observe that \(Q\subset Q_N\)  for some \(N\in {\mathbb {N}}\) . When \(Q\)  contains at most 4 points of \(E,\)  it is straightforward to see that the distance \(d_E\)  satisfies (REF ) for \(Q \)  and with some constant \(C_1\)  only depending on \(\alpha \)  and \(p.\)  This includes the case where \(Q \)  is contained in \(Q_1.\)
We prove by induction on \(N\)  that \(d_E\)  satisfies (REF ) for every interval \(Q \subset Q_N\)  with the constant \(C=\max \lbrace 12^p \widehat{C}, C_1\rbrace \) , where \(\widehat{C}\)  is the constant in Lemma REF . The case \(N=1\)  has already been proved since \(C\ge C_1\) . Hence, we assume that the claim holds for all \(n=1, \ldots , N-1,\) 
and we need to verify the claim for all intervals \(Q\)  contained in \(Q_N.\)
The case where \(Q \subset Q_{N-1}\)  follows from the induction hypothesis. Thus we may and do assume that \(Q\)  is not contained in \(Q_{N-1}.\)  We do a case study.
(i): \(Q\)  is contained in one of the intervals \(Q_{N-1}^1\) , \(Q_{N-1}^2.\)  In the first case, the cube \(Q\subset Q_{N-1}^1\)  can be written as \(Q=t_N Q^*\) , where \(Q^*\)  is a translation of an interval \(\widehat{Q}\)  contained in \(Q_{N-1}.\)  Then \(\vert Q \vert = t_N \vert \widehat{Q} \vert \)  and \(\int _Q d_E^\beta = t_N^{1+\beta } \int _{\widehat{Q}} d_E^\beta \)  for every \(\beta >-1.\)  This gives
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }} d_E^{-\alpha }\left( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }} d_E^{\frac{\alpha }{p-1}} \right)^{p-1} = \left( t_N^{-\alpha } \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}\widehat{Q}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{\widehat{Q}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{\widehat{Q}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{\widehat{Q}}} d_E^{-\alpha } \right) \left( t_N^{\frac{\alpha }{p-1}} \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}\widehat{Q}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{\widehat{Q}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{\widehat{Q}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{\widehat{Q}}} d_E^{\frac{\alpha }{p-1}} \right)^{p-1} =\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}\widehat{Q}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{\widehat{Q}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{\widehat{Q}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{\widehat{Q}}} d_E^{-\alpha }\left( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}\widehat{Q}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{\widehat{Q}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{\widehat{Q}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{\widehat{Q}}} d_E^{\frac{\alpha }{p-1}} \right)^{p-1} \le C,\) 
where the last inequality holds by the induction hypothesis. In the second case we have \(Q\subset Q_{N-1}^2,\)  and inequality (REF ) follows from the induction hypothesis since \(Q\)  is now translation of an interval \(\widehat{Q}\)  contained in \(Q_{N-1}\) .
(ii): \(Q\)  intersects both \(Q_{N-1}\)  and \(Q_{N-1}^2.\)  This implies that \(Q\)  contains \(Q_{N-1}^1,\)  and so
\(\vert Q \vert \ge |Q_{N-1}^1|= t_N \vert Q_{N-1} \vert = \frac{t_N}{2+t_N} \vert Q_N \vert \ge \frac{1}{6} \vert Q_N \vert .\) 
Using this estimate together with Lemma REF , we obtain
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }} d_E^{-\alpha }\left( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }} d_E^{\frac{\alpha }{p-1}} \right)^{p-1} &\le \left( \frac{6}{ \vert Q_N \vert }\int _{Q } d_E^{-\alpha } \right)\left( \frac{6}{\vert Q_N \vert }\int _{Q } d_E^{\frac{\alpha }{p-1}} \right)^{p-1} \\&\le 6^p \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}} d_E^{-\alpha }\left( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}} d_E^{\frac{\alpha }{p-1}} \right)^{p-1} \le 6^p \widehat{C}.\) 
(iii): \(Q\)  contains one of the intervals \(Q_{N-1},\)  \(Q_{N-1}^1\) , \(Q_{N-1}^2.\)  In this case
\(|Q| \ge t_N \vert Q_{N-1} \vert \ge \frac{1}{6} \vert Q_N \vert .\)  Using that \(Q \subset Q_N,\)  the desired estimate follows as in the case (ii).
(iv): Assume that \(Q \cap Q_{N-1} \ne \emptyset \ne Q \cap Q_{N-1}^1\)  but \(Q \cap Q_{N-1}^2=\emptyset .\)  By the construction of \(Q_{N-1}\) , we can find \(m\in \lbrace -1, 0, \ldots , N-2 \rbrace \)  so that \( Q_m^* \subset Q \cap Q_{N-1} \subset Q_{m+1}^* ,\)  where \(Q_m^*\)  and \(Q_{m+1}^*\)  are translations of \(Q_m\)  and \(Q_{m+1}\)  respectively, and we use the notation \(Q_{-1}^*=\emptyset .\)  This implies \(\vert Q \cap Q_{N-1} \vert \ge \vert Q_{m} \vert .\) 
Similarly, by the construction of \(Q_{N-1}^1,\)  there exists \(n \in \lbrace -1, 0, \ldots , N-2 \rbrace \)  so that \(t_N Q_n^* \subset Q \cap Q_{N-1}^1 \subset t_N Q_{n+1}^* ,\)  where \(Q_n^*\)  and \(Q_{n+1}^*\)  are translations of \(Q_n\)  and \(Q_{n+1}\) , respectively, and so \(\vert Q \cap Q_{N-1}^1 \vert \ge t_N \vert Q_n \vert .\)  Now define \(M=\max \lbrace m,n\rbrace .\)  If \(M=-1,\)  then \(Q\)  intersects at most 2 edges of \(E,\)  and the desired estimate follows with the constant \(C_1\)  from the beginning of the proof. If \(M \ge 0,\)  then we have \(Q \cap Q_{N-1} \subset Q_{M+1}^*\)  and \(Q \cap Q_{N-1}^1 \subset t_N Q_{M+1}^*,\)  and so
\(\int _Q d_E^\beta \le \int _{Q_{M+1} }d_E^\beta + t_N^{1+\beta }\int _{Q_{M+1} }d_E^\beta = \left( 1+ t_N^{1+\beta } \right) \int _{Q_{M+1} }d_E^\beta \le 2 \int _{Q_{M+1} }d_E^\beta ,\) 
for every \(\beta >-1.\)  On the other hand,
\(\vert Q \vert &= \vert Q \cap Q_{N-1} \vert + \vert Q \cap Q_{N-1}^1 \vert \ge \vert Q_{m} \vert + t_N \vert Q_{n} \vert \\& \ge t_N \vert Q_{M} \vert = \frac{t_N}{2+ t_{M+1}} \vert Q_{M+1} \vert \ge \frac{|Q_{M+1}|}{6}.\) 

This leads us to
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }} d_E^{-\alpha }\left( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }} d_E^{\frac{\alpha }{p-1}} \right)^{p-1}\le 12^p \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_{M+1}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_{M+1}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_{M+1}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_{M+1}}} d_E^{-\alpha }\biggl ( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_{M+1}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_{M+1}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_{M+1}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_{M+1}}} d_E^{\frac{\alpha }{p-1}} \biggr )^{p-1} \le 12^p \widehat{C},\) 
where the last inequality follows from Lemma REF .
(v): Assume that \(Q \cap Q_{N-1}^1 \ne \emptyset \ne Q \cap Q_{N-1}^2\)  but \(Q \cap Q_{N-1}=\emptyset .\)  Recall that \(Q_{N-1}^2\)  is a translation of \(Q_{N-1}\)  that contains, from left to right, translated copies \(Q_0^* \subset Q_1^* \subset \cdots \subset Q_{N-2}^* \subset Q_{N-1}^*\)  of \(Q_0\subset Q_1 \subset \cdots \subset Q_{N-2} \subset Q_{N-1}\) , respectively. In addition, \(Q_{N-1}^1\)  contains, from right to left, translated copies \(t_N Q_{N-1}^* \supset t_N Q_{N-2}^* \supset \cdots \supset t_N Q_1^* \supset t_N Q_0^* \)  of \(Q_{N-1} \supset Q_{N-2} \supset \cdots \supset Q_1 \supset Q_0\)  dilated by \(t_N.\)  Now, the argument is identical to the case (iv).
Lemma 8.7 
Let \(0<\alpha <1\)  and \(1<p<\infty \) , and let \(C=C(\alpha ,p)\)  be the constant in Lemma REF .
Then
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }} d_E(x)^{-\alpha }\,dx \left( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }} d_E(x)^{\frac{\alpha }{p-1}}\,dx \right)^{p-1} \le 2^p C\) 
for every cube \(Q \subset {\mathbb {R}}\) , and so \(d_E^{-\alpha } \in A_p.\)

Given a cube \(Q \subset {\mathbb {R}}\) , we write \(Q^+=Q \cap [0,+ \infty )\)  and \(Q^-=Q \cap (-\infty , 0].\) 
Let \(Q^*\)  be the largest of the cubes \(Q^+\)  and \(-Q^-\) , that is, \(Q^*\in \lbrace Q^+, -Q^- \rbrace \) 
and \(Q^+ \cup -Q^-\subset Q^*\) . Here
\(-Q^{-}\)  denotes the reflection of \(Q^{-}\)  with respect to the origin.
Because \(E\)  is symmetric with respect to the origin, we can write
\(\int _Q d_E^{-\alpha } = \int _{Q^+} d_E^{-\alpha } + \int _{Q^-} d_E^{-\alpha } = \int _{Q^+} d_E^{-\alpha } + \int _{-Q^-} d_E^{-\alpha } \le 2 \int _{Q^*} d_E^{-\alpha }.\) 
The same argument shows that \(\int _Q d_E^{\frac{\alpha }{p-1}} \le 2 \int _{Q^*} d_E^{\frac{\alpha }{p-1}}.\)  Because \(\vert Q\vert \ge \vert Q^*\vert \)  and \(Q^*\)  is contained in \([0,\infty ),\)  we can use Lemma REF  to conclude
that
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }} d_E^{-\alpha }\left( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }} d_E^{\frac{\alpha }{p-1}} \right)^{p-1} \le 2^p \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q^*}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q^*}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q^*}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q^*}} d_E^{-\alpha }\left( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q^*}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q^*}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q^*}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q^*}} d_E^{\frac{\alpha }{p-1}} \right)^{p-1} \le 2^p C. \) 
